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Quiz 3-1

Quiz 3-1. This data can be modeled using an exponential equation . Find ‘a’ and ‘b’. Where does cross the y-axis ?. 3. Is g(x) an exponential growth or decay function?. 4. Convert to exponential notation:. 5. Convert to logarithmic notation:. 3.2.

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Quiz 3-1

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  1. Quiz 3-1 • This data can be modeled using an • exponential equation  Find ‘a’ and ‘b’ • Where does cross the y-axis ? 3. Is g(x) an exponential growth or decay function? 4. Convert to exponential notation: 5. Convert to logarithmic notation:

  2. 3.2 Exponential and Logistic Modeling

  3. What you’ll learn about • Constant Percentage Rate and Exponential Functions • Exponential Growth and Decay Models • Using Regression to Model Population • Other Logistic Models … and why Exponential functions model unrestricted growth (money) and decay (radioactive material); Logistic functions model restricted growth, (spread of disease, populations and rumors)

  4. Factoring Your turn: Factor the following 1. f(x) = 3 + 3x 2. g(y) = 5 + 5y

  5. Constant Percentage Rate A population is changing at a constant percentagerate r, where r is the percent rate (in decimal form). Time (years) Population “initial population” 0 1 Your turn: 3. Factor P(1) 2 Your turn: 4. Factor P(2) Your turn: 5. Write P(2) in terms of P(0) only.

  6. Constant Percentage Rate Time (years) Population 0 “initial population” 1 2 Your turn: 6. What do you think P(3) will be? 3 4 t

  7. Exponential Population Model If a population is changing at a constant percentage rate ‘r’ each year, then: is the population as a function of time.

  8. Finding Growth and Decay Rates Is the following population model an exponential growth or decay function? Find the constant percentage growth (decay) rate. ‘r’ > 0, therefore this is exponential growth. ‘r’ = 0.0136 or 1.36%

  9. Finding an Exponential Function Determine the exponential function with initial value = 10, increasing at a rate of 5% per year. ‘r’ = 0.05 or

  10. Your Turn: • The population of “Smallville” in the year • 1890 was 6250. Assume the population • increased at a rate of 2.75% per year. • What is the population in 1915 ?

  11. Modeling Bacteria Growth Suppose a culture of 100 bacteria cells are put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350,000. P(0) = 100 P(t) = 350000 Doubles with Every time interval P(1) = 2*P(0)

  12. Solving an Exponential Equation Your calculator doesn’t have base 2 (it might in some of the catalog of functions) Change of Base Formula: t = 11 hours, 46 minutes

  13. Your Turn: The population of “Smallville” in the year 1890 was 6250. Assume the population increased at a rate of 2.75% per year. 8. When did the population reach 50,000 ?

  14. Exponential Regression Stat p/b  gives lists Enter the data: Let L1 be years since initial value Let L2 be population Stat p/b  calc p/b scroll down to exponential regression “ExpReg” displayed: enter the lists: “L1,L2” The calculator will display the values for ‘a’ and ‘b’.

  15. Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003. (Don’t enter the 2003 value). Let P(t) = population, “t” years after 1900. Enter the data into your calculator and use exponential regression to determine the model (equation).

  16. Modeling U.S. Population Using Exponential Regression Your turn: • 9. What is your equation? • What is your predicted population • in 2003 ? • Why isn’t your predicted value • the same as the actual value of • 290.8 million?

  17. Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population. We must use Logistic function if the growth is limited !!!

  18. Modeling a Rumor Roy High School has about 1500 students. 5 students start a rumor, which spreads logistically so that Models the number of students who have heard the rumor by the end of ‘t’ days, where ‘t’ = 0 is the day the rumor began to spread. How many students have heard the rumor by the end of day ‘0’ ? How long does it take for 1000 students to have heard the rumor ?

  19. Rumors at RHS How many students have heard the rumor by the end of day ‘0’ ? How long does it take for 1000 students to have heard the rumor ? Your turn: 12. “t” = ? (days)

  20. HOMEWORK

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